4-x-x-260-find-the-possible-values-of-x-

Question Number 205238 by necx122 last updated on 13/Mar/24

4^{x} + x = 260

f i n d t h e p o s s i b l e v a l u e s o f x

Commented by Ghisom last updated on 13/Mar/24

generally

b^{x} = a x + c

let x = - \frac{t}{\ln b} - \frac{c}{a}

e^{t} t = - \frac{\ln b}{b^{c / a} a}

t = W (- \frac{\ln b}{b^{c / a} a})

x = - \frac{c}{a} - \frac{1}{\ln b} W (- \frac{\ln b}{b^{c / a} a})

Answered by A5T last updated on 13/Mar/24

4^{4} = 2^{8} = 256 = 260 - 4 \Rightarrow x = 4

w h e n x > 4, 4^{x} + x > 256 + 4 = 260

w h e n x < 4, 4^{x} + x < 260 \Rightarrow x = 4

Commented by necx122 last updated on 13/Mar/24

can it be solved analytically? I know of Lambert W function but I don't know how to apply it here.

Answered by mr W last updated on 13/Mar/24

4^{x} = 260 - x

4^{x - 260} \times 4^{260} = 260 - x

(260 - x) 4^{260 - x} = 4^{260}

(260 - x) e^{(260 - x) \ln 4} = 4^{260}

(260 - x) \ln 4 e^{(260 - x) \ln 4} = 4^{260} \times \ln 4

\Rightarrow (260 - x) \ln 4 = W (4^{260} \times \ln 4)

\Rightarrow x = 260 - \frac{W (4^{260} \times \ln 4)}{\ln 4} = 4

Commented by necx122 last updated on 13/Mar/24

This is so clear and understandable. Thank you sir.

Commented by necx122 last updated on 13/Mar/24

M e a n w h i l e, i s t h e L a m b e r t W f u n c t i o n

s u p p o s e d t o g i v e u s o t h e r v a l u e s ?

H o w d o w e c a l c u l a t e f o r o t h e r s ?

W h a t c a l c u l a t o r s d o w e u s e f o r

c o m p u t i n g t h e l a m b a r t W f u n c t i o n .

Commented by mr W last updated on 13/Mar/24

i u s e w o l f r a m a l p h a

Commented by mr W last updated on 13/Mar/24

Commented by mr W last updated on 13/Mar/24

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